Homography

Homography is a concept in the mathematical science of geometry. It is defined as a relation between two figures, such that any given point in one figure corresponds to one and only one point in the other, and vice versa.

Computer Vision Applications

In the field of computer vision, a homography is defined in 2 dimensional space as a mapping between a point on a ground plane as seen from one camera, to the same point on the ground plane as seen from a second camera. This has many practical applications, most notably it provides a method for compositing 2D or 3D objects into an image or video with the correct pose. The homography matrix is sometimes known as a homograph, a term which has a different meaning in linguistics.

3D plane to plane equation

We have two cameras a and b, looking at points $P\_i$ in a plane.

Passing the projections of $P\_i$ from $\{\}^bp\_i$ in b to a point $\{\}^ap\_i$ in a:

$\{\}^ap\_i\; =\; K\_a\; cdot\; H\_\{ba\}\; cdot\; K\_b^\{-1\}\; cdot\; \{\}^bp\_i$

where $H\_\{ba\}$ is

$H\_\{ba\}\; =\; R\; -\; frac\{t\; n^T\}\{d\}$

$R$ is the rotation matrix by which "b" is rotated in relation to "a"; "t" is the translation vector from a to b; $n$ and $d$ are the normal vector of the plane and the distance to the plane respectively.

$K\_a$ and $K\_b$ are the cameras' intrinsic parameter matrices.

The figure shows camera b looking at the plane at distance d.

Mathematical definition

Homogeneous coordinates are used, because matrix multiplication cannot directly perform the division required for perspective projection.

Given::Then::$p^\{prime\}\_\{b\}\; =\; mathbf\{H\}\_\{ab\}p\_\{a\}$where::$mathbf\{H\}\_\{ba\}\; =\; mathbf\{H\}\_\{ab\}^\{-1\}.$Also::

Affine homography

When the image region in which the homography is computed is small or the image hasbeen acquired with a large focal length, an "affine homography" is a more appropriatemodel of image displacements. An affine homography is a special type of a generalhomography whose last row is fixed to $h\_\{31\}=h\_\{32\}=0,\; ;\; h\_\{33\}=1$.

See also

*Epipolar geometry

External links

*M. Lourakis' [http://www.ics.forth.gr/%7elourakis/homest/ homest] is a GPL C/C++ library for robust, non-linear (based on the Levenberg-Marquardt algorithm) homography estimation from matched point pairs. homest can estimate fully projective and affine homographies.
* [http://www.robots.ox.ac.uk/%7Evgg/presentations/bmvc97/criminispaper/node3.html Computing the plane to plane homography]
* [http://www.cse.iitd.ac.in/~suban/vision/geometry/node24.html How to compute a homography]
* [http://www.robots.ox.ac.uk/%7Evgg/hzbook/code/ MATLAB Functions for Multiple View Geometry] Matlab functions for calculating a homography and the fundamental matrix